A Proposed Approach for Solving Rough Bi-Level. Programming Problems by Genetic Algorithm

Transcription

1 Int J Contemp Math Sciences, Vol 6, 0, no 0, A Proposed Approach or Solving Rough Bi-Level Programming Problems by Genetic Algorithm M S Osman Department o Basic Science, Higher Technological Institute Ramadan 0 th city, Egypt W Abd El-Wahed Department o Operations Researches and Decision Making aculty o Computers and Inormation, El-Menouia University Shebin El-Kom, Egypt Mervat M K El Shaei Department o Mathematics, aculty o Science Helwan University, Cairo, Egypt mervat_elshaei@yahoocom Hanaa B Abd El Wahab Department o Basic Science, Special Studies Academy, Cairo, Egypt Abstract We present a new ramework to hybridize the rough set theory with the bi-level programming problem, called Rough Bi-level Programming Problems (RBLPPsThis paper studies and designs a genetic algorithm (GA or solving (RBLPPs by constructing the itness unction o the upper level programming problems based on the deinition o through easible degree inally, a numerical example will be introduced to show the proposed method Keywords: Rough set, Rough Bi- level linear programming problems, Rough optimality, Rough easibility, Rough easible degree; itness unction; genetic algorithm

2 454 M S Osman et al Introduction Since it was pioneered by Pawalk in mid 980 s and [4] rough set theory has become a hot topic o great interest to researchers in several ields and has been applied to many domains such as pattern recognition, data mining, artiicial intelligence, image processing,machine learning, and medical application This new approach proved to be useul in many applications such as optimization theory or mathematical programming problems (MPPs in the crisp orm, the aim is to maximize or minimize an objective unction over certain set o easible solutions But in many practical situations, the decision maker may not be in a position to speciy the objective and/or the easible set precisely but rather can speciy them in a rough sense In such situations, it is desirable to use some rough programming type o modeling so as to provide more lexibility to the decision maker Towards this objective, we present a new ramework to hybridize the rough set theory with the bi-level programming problem, called Rough Bilevel Programming Problems (RBLPPs Since the roughness may appear in a Bi-level Programming Problems in many ways (eg the easible set may be rough and/or the goals may be rough, the deinition o rough Bi-level Programming Problems is not unique This leads us to propose a new classiication and characterization o the rough Bi-level Programming Problems (RBLPPs, according to the place o roughness in the problem We classiied the RBLPPs into the ollowing classes: Bi-level programming problems with rough easible set, crisp objective unction o the upper level and crisp objective unction o the lower Bi-level programming problems with rough easible set, crisp objective unction o the upper level and rough objective unction o the lower Bi-level programming problems with rough easible set, rough objective unction o the upper level and crisp objective unction o the lower 4 Bi-level programming problems with crisp easible set and rough objective unction o the upper level and crisp objective unction o the lower 5 Bi-level programming problems with crisp easible set and crisp objective unction o the upper level and rough objective unction o the lower 6 Bi-level programming problems with crisp easible set and rough objective unction o the upper level and rough objective unction o the lower 7 Bi-level programming problems with rough easible set and rough objective unction o the upper level and rough objective unction o the lower New deinitions concerning rough optimal sets, rough optimal value, rough global optimality and rough easibility were also proposed and discussed This paper studies and designs a genetic algorithm (GA o (RBLPPs by constructing the itness unction o the upper level programming problems based on the deinition o the rough easible degree

3 Rough bi-level programming problems 455 Rough set and approximation space Rough set theory has been proven to be an excellent mathematical tool dealing with vague description o objects [4] A undamental assumption in rough set theory is that any object rom a universe is perceived trough available inormation, and such inormation may not be suicient to characterize the object exactly Pawlak has proposed rough set methodology as a new approach in handling classiicatory analysis o vague concepts [4] In this methodology any vague concept is characterized by a pair o precise concepts called the lower and the upper approximations Rough set theory is based on equivalence relations describing partitions made o classes o indiscernible objects Let U be a non-empty inite set o objects, called the universe, and E U U be an equivalence relation on U The ordered pair A ( U, E is called an approximation space generated by E on U The equivalence relation E generates a partition U / E { Y, Y,, Y m where Y, Y,, Ym are the equivalence classes (also called elementary sets or granules generated by E, represent elementary portion o knowledge we are able to perceive due to E o the approximation space A In rough set theory, any subset M U is described by the elementary sets o A, and the two sets E ( M U { Y i U / E Y i M E ( M U { Y U / E Y I M φ i i are called the lower and the upper approximations o M, respectively Thereore, E ( M M E ( M The dierence between the upper and the lower approximations is called the boundary o M and is denoted by BN E (M E ( M E ( M The set M is called exact in A i BN E (M φ ; otherwise the set M is inexact (rough in A [5, 6, 8, 9] As we can see rom the deinition approximations are expressed in terms o granules o knowledge The lower approximation o a set is union o all granules which are entirely included in the set; the upper approximation - the union o all granules which have non empty intersection with the set; the boundary region o the set is the dierence between the upper and lower approximation [4] This deinition is clearly depicted in igure Classes o Rough Bi-level Programming Problems (RBLPPs be stated as: ( where x solve The most typical Bi-level Programming Problems (BLPPs can max ( x, x x

4 456 M S Osman et al max x ( x x, subject to x ( x, x M where and are called the objective unctions o the upper and lower level DM, and M is called the easible set o the problem In the above ormulation, it is assumed that all entries o, and M are deined in the crisp sense, and max is a strict imperative However, in many practical situations it may not be reasonable to require that the easible set or the objective unctions in bi-level programming problems be speciied in precise crisp terms In such situations, it is desirable to use some type o rough modeling and this leads to the concept o rough bi-level programming problems When decision is to be made in a rough environment, many possible modiications o the above bi-level programming model exist Thus, rough bi-level programming models are not uniquely deined as it will very much depend upon the type o roughness and its speciication as prescribed by the decision maker Thereore, the rough bi-level programming problems can be broadly classiied as: st Class: Bi-level programming problems with rough easible set, crisp objective unction o the upper level and crisp objective unction o the lower nd Class: Bi-level programming problems with rough easible set, crisp objective unction o the upper level and rough objective unction o the lower rd Class: Bi-level programming problems with rough easible set, rough objective unction o the upper level and crisp objective unction o the lower th 4 Class: Bi-level programming problems with crisp easible set and rough objective unction o the upper level and crisp objective unction o the lower th 5 Class: Bi-level programming problems with crisp easible set and crisp objective unction o the upper level and rough objective unction o the lower th 6 Class: Bi-level programming problems with crisp easible set and rough objective unction o the upper level and rough objective unction o the lower th 7 Class: Bi-level programming problems with rough easible set and rough objective unction o the upper level and rough objective unction o the lower In RBLPPs, wherever roughness exists, new concepts like rough easibility and rough global optimality come in the ront o our interest The rough easibility arises only in the st, nd, rd and 7 th classes, where solutions have dierent degrees o easibility (surely-easible, possible-easible, and surely-not easible On the other hand, the rough global optimality arises in all classes o the RBLPPs where solutions have dierent degrees o global optimality (surelyglobal optimal, possible- global optimal, and surely-not global optimal As a result o these new concepts, the optimal value o the objectives and the optimal set o the problem are deined in rough sense (

5 Rough bi-level programming problems 457 Deinition : In RBLPPs, the optimal value o the objective unction o the upper level DM is a rough real number, that is determined roughly by lower and upper bounds denoted by,, respectively Remark : I then the optimal value is exact, otherwise is rough Also, the single optimal set o the crisp bi-level programming problem is replaced by our optimal sets covering all possible degrees o easibility and optimality See table, igure Deinition : The set o all surely-easible, surely- global optimal solutions is denoted by O Deinition : The set o all surely-easible, possibly-global optimal solutions is denoted by O sp ss

6 458 M S Osman et al Deinition 4: The set o all possibly -easible, surely -global optimal solutions is denoted by O ps Optimality Possibly Surly O pp O Possibly ps O Surly sp O ss easibi lity Table : Possible degrees o easibility and optimality or problem (RBLPPs Deinition 5: The set o all possibly -easible, possibly -global optimal solutions is denoted by O pp 4 st Class RBLPPs Suppose that A ( U, E is an approximation space generated by equivalence relation E on an universe U A rough bi-level programming problem o the st class takes the ollowing orm: max ( x, x ( x where x solve max ( x, x (4 x subject to E ( M M E ( M where M U is a rough set in the approximation space A ( U, E representing the easible set o the problem The sets E ( M M and E ( M M represent the notion o rough easibility o problem ( - (4, where M is called the set o all possibly easible solutions and M is called the set o all surely-easible solutions On the other hand U M is called the set o all surely-not easible solutions [] Proposition : In problem ( - (4, the lower and upper bounds o the optimal objective value or the upper level problem are given by sup{ a, b sup{ a, c where a max ( x, x ( x x, M

7 Rough bi-level programming problems 459 b sup { min ( x, x c ( x x U Y U / E ( x, x Y Y M BN max ( x, x, M BN Deinition 6: A solution ( x,x M is surely -global optimal solution i ( x, x Deinition 7: A solution ( x,x M is possibly -global optimal solution i ( x, x Deinition 8: A solution ( x,x M is surely -not global optimal solution i ( x, x < Deinition 9: The optimal sets o the st class RBLPP or the upper level problem are deined as: O ss x, x M ( x, x O sp { ( { ( x, x M ( x, x { ( x, x M ( x, x ( x, x M ( x, x O ps O pp { Proposition : The lower and upper bounds o the optimal objective value or the lower level problem are given by sup{ a, b sup{ a, c where a max ( x, x b ( x, x M sup c U Y U / E ( x, x Y Y M BN max ( x, x ( x, x M BN min ( x, x Deinition 0: A solution ( x, x M is surely -global optimal solution i ( x, x Deinition : A solution ( x, x M is possibly -global optimal solution i ( x, x Deinition : A solution ( x, x M is surely not -global optimal solution i ( x, x < Deinition : The optimal sets o the st class RBLPP or the lower level problem are deined as: { ( x, x M ( x, x O ss

8 460 M S Osman et al O sp { ( x, x M ( x, x ( x, x M ( x, x ( x, x M ( x, x { { O ps O pp 5 Design o GA or RBLPPs The basic idea or solving RBLPP by GA is: irstly, choose the initial population satisying the constraints (by using the above deinitions and proposition,, the lower-level decision maker makes the corresponding rough optimal reaction and evaluate the individuals according to the itness unction constructed by the rough easible degree, until the optimal solution searched by the genetic operation over and over 6 Design o the itness unction To solve the problem RBLPPs by GA, the deinition o the easible degree is irstly introduced and the itness unction is constructed to solve the problem by GA Let d denote the large enough penalty interval o the rough easible region or each ( x, x X Deinition 4: Let θ [0,] denotes the rough easible degree o satisying the rough easible region, and describe it by the ollowing unction:, i x Y ( x 0 x Y ( x θ, i 0 < x Y ( x d (5 d 0, i x Y ( x > d where denotes the norm and Y ( x denote the rough optimal solution o lower level problem urthers, the itness unction o the GA can be stated as: eval ( vk ( ( x, x min θ where min is the rough minimal value o ( x, x on X

9 Rough bi-level programming problems 46 7 The algorithm Step Initialization, give the population scale M, the maximal iteration generation MAXGEN, and let the generation t 0 [7] Step The initial population, M individuals are randomly generated in X, making up the nitial population Step Evaluation the rough minimal value o upper level problem ( min and the rough optimal solution o the lower level ( Y ( x by using proposition, respectively Step 4 Computation o the itness unction Evaluate the itness value o the population according to ormula (5 Step 5 Selection, Select the individual by roulette wheel selection operator [] Step 6 Crossover, in this step, irst, a random number P c [0,] is generated This number is the percentage o the population on which the crossover is perormed Then, two individuals are selected randomly rom the population as parents Children are generated using the ollowing procedure: Random integer c is generated in the interval [, l ], where l is the number o components o an individual The c irst components o the children are the same components as respective parents (ie the irst child rom the irst parent and the second child rom the second parent The remaining components are selected according to the ollowing rules: (i The ( c + i th component o the irst child is replaced by the ( l i + th component o the second parent (ori,,, l c (ii The ( c + i th component o the second child is replaced by the ( l i + th component o the irst parent (ori,,, l c or example we assume c 5, we obtained the ollowing children Parents children Note that the proposed operator generates individuals with more variety in comparison with the standard operator, because this operator can generate dierent children rom similar parents, where standard operators cannot [] Step 7 Mutation, In this step, irst, a random number Pm [0,] is generated This number is the percentage o the population on which the mutation perormed Then one individual is selected randomly rom the population An integer random number u is generated in the interval [, l ], where l is the length o the individual or generating the new individual, the u th component is changed to 0, i it was initially and to i it was initially 0 [] Step 8 Terminations, Jude the condition o the termination When t is larger than the maximal iteration number, stop the GA and output the rough optimal solution Otherwise, let t t +, turn to Step

10 46 M S Osman et al 8 An Example Let U be a universal set deined as x ( x, x { R x + 9 U x and let K be a polytope generated by the ollowing closed halplanes h x + x 0, h x x 0 h x x + 0, h4 x + x + 0 Suppose that E is an equivalence relation on U such that U / E E, E E {, { x U : x is an interior point o polytope K { x U : x is a boundary point o polytope K { x U : x is an exterior point o polytope K E E E Consider the ollowing st Class RBLPPs max ( x, x x x x M where x solve max ( x, x x M ( 5 x + x subject to M E U E, M E U E U E where M is a rough easible region in the approximation space A ( U, E and M, M are the lower and the upper approximation o M ; respectively Also, the boundary region o M is given by M BN E The solution By using the above genetic algorithm we ound the ollowing results: Give the population scale M 00, the maximal iteration generation MAXGEN 0, and let the generation t 0 Computation o the itness unction Evaluate the itness value o the population according to ormula (5 as: eval( vk ( ( x 5 x 044 θ To evaluate min, we use proposition and the given deinitions inding the rough minimal value o the upper level problem min x x 5 ( x + x sup{ a, b a max ( x, x ( 0, Where, ( Q ( x, x in ( x, x M b sup ( x, x (0000, 0 65 E U { min ( x, x Y U / E, ( x, x Y Y M 044 BN min ( x, x (0000, 0 65 ( x x, E (6 (7

11 Rough bi-level programming problems 46 Q sup{ a, c Q sup ( x, x ( x, x E c max ( 0000, 0 65 ( x, x max ( x, x M ( x, x BN E ( x, x ( 0000, min 0 44 inding the rough optimal value o the upper level problem : sup{ a, b a max x, x (,0 05 Q ( x, x in ( x, x M b sup ( ( x, x (,0 0 5 E U { min ( x, x Y U / E, ( x, x Y Y M BN 05 Q sup{ a, c Q sup x, x ( x, x E c max ( x, x M ( BN (5, 0 0 ( x, x max ( x, x E min ( x x E, ( x, x (,0 0 5 ( x, x (5, [ 05, 0] inding the rough optimal sets or the upper level problem: O ss { ( x, x E U E : ( x, x 0 φ O sp { ( x, x E U E : ( x, x 0 5 {(,0 O ps { ( x, x E U E U E : ( x, x 0 {( 5,0 O PP {( x, x E U E U E : ( x, x 05 {( x, x : ( x 5 + x 05 But or evaluateθ, we ind Y ( x by using proposition such that: inding the rough optimal value o the lower level DM Y ( x x + or ixed x sup{ a, b a max ( x, x (, 4 ( x, x [,] Q in ( x, x (, ( x, x E b sup ( U min ( x, x min ( x, x (, Y U / E, x, x Y ( x, x E 4 Y M BN

12 464 M S Osman et al sup{ a, c Q sup ( x, x E ( x, x (, 5 c max ( x, x max ( x, x M ( x, x 5 [ 4, 5 ] BN E ( x, x (, 5 inding the rough optimal sets or the lower level O ss x, x E U E : x, x 5 { ( ( φ { ( x, x E E : ( x, x 4 { { ( x, x E U E E : ( x, x 5 { { ( x x E U E U E : ( x, 4 [,] O sp U O ps U O x PP, then Y ( x 4 and d 8 The operators o the genetic algorithm [, 7] are applied such that P 0 7, P 0 c m Obtain the optimal solution ( x, x, θ ( 5,, 055 Elapsed time is 445 minutes 9 Conclusions This paper proposes a new ormulation, classiication and deinition o the rough bi-level programming problems Only the st class o the RBLPPs is deined and its optimal sets are characterized in this work Also, it designs GA or solving RBLPPs o which the rough optimal solution o the lower-level problem is dependent on the upper-level problem Reerences [] SR Hejazi, A Memariani, G Jahanshahloo, M M Sepehri, Linear Bi-Level Programming Solution by Genetic Algorithm, Computers and Operations Research, 9 (00, 9-95 [] Z Michalewicz,Genetic Algorithms +Data StructuresEvolution Programs, Springer, New York, 99 [] MS Osman, E Lashein, EA Youness, TEM Atteya, Rough Mathematical Programming, Optimization A Journal o Mathematical Programming and Operations Research, 58 (009,-8

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